9.13 三角形的未知维度

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  三角形的未知尺寸

  

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  Bill and Takeen are running backs for the high school football team. They need to be in a split running back formation, which is a triangular pattern, and stay on a line 5 yards behind the quarterback. The area of the triangle should be approximately 20 square yards. How far apart should Bill and Takeen stand from one another?
  ::Bill和Takeen正在为高中足球队跑后跑。他们需要处于一个分裂的后跑阵形(三角形),在四分卫后面的5码线上。三角区应该大约为20平方码。 Bill和Takeen应该相互站在一起多远?

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  In this concept, you will learn how to find the unknown dimensions in a triangle.
  ::在这个概念中,你会学会如何在三角形中找到未知的维度。

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  Finding Unknown Dimensions in a Triangle
  ::在三角中查找未知尺寸

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  The area of a triangle can be found by multiplying the base times the height and then dividing the product by 2.
  ::三角形区域可以通过乘以基数的高度乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以乘以。

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  $$\begin{array}{r}A=\frac{bh}{2}\end{array}$$

  
  ::A=bh2 阿=bh2

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  This equation can also be used if you are given the area and asked to find either the base or the height.
  ::如果给您给定区域并要求您找到基数或高度,此方程式也可以使用。

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  Here’s an example:
  ::举例来说:

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  A triangle has an area of $\begin{array}{r}44m^2\end{array}$ . The base of the triangle is 8 m. What is its height?
  ::三角形的面积为44平方米,三角形的底部是8米。它的高度是多少?

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  First, write the formula and substitute the values that are given.
  ::首先,写入公式并替换给定的值。

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  $$\begin{array}{r}\begin{array}{rcl}A& =& \frac{bh}{2}\\ 44& =& \frac{8h}{2}\end{array}\end{array}$$

  
  ::A=bh244=8h2

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  Next, begin isolating your unknown by multiplying both sides of the equation by 2.
  ::接下来,通过将方程的两边乘以2 来孤立未知的你

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  $$\begin{array}{r}\begin{array}{rcl}44\times 2& =& \frac{8h}{2}\times 2\\ 88& =& 8h\end{array}\end{array}$$

  
  ::44x2=8h2x288=8h

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  Then, divide both sides by 8.
  ::然后,将双方除以8。

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  $$\begin{array}{r}\begin{array}{rcl}88& =& 8h\\ 11& =& h\end{array}\end{array}$$

  
  ::88=8h11=h

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  The answer is 11 m. The height of the triangle is 11 m.
  ::答案是11米 三角形的高度是11米

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  You can check your answer by placing all the dimensions back into the equation.
  ::您可以通过将所有维度都放回方程中来检查您的答案 。

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  $$\begin{array}{r}\begin{array}{rcl}A& =& \frac{bh}{2}\\ 44& =& \frac{(8)(11)}{2}\\ 44& =& \frac{88}{2}\\ 44& =& 44\end{array}\end{array}$$

  
  ::A=bh244=(8)(11)(11)244=88244=44

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were given a problem about  the running backs, Bill and Takeen.
  ::早些时候,你遇到了一个问题 关于跑后退,比尔和Takeen。

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  They needed to know how far apart they should stand from one another to maintain a 20 square foot triangular pattern with the quarterback who is on a line 5 yards in front of them.
  ::他们需要知道他们应该站在一起多远 才能保持20平方英尺的三角模式 与站在他们前面5码线上的四分卫保持距离

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  First, draw a picture and label it.
  ::首先,画一张画,贴上标签。

  

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  Next, fill in the formula with the values that are given.
  ::下一步,用给定的值填入公式。

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  $$\begin{array}{r}\begin{array}{rcl}A& =& \frac{bh}{2}\\ 20& =& \frac{(b)(5)}{2}\end{array}\end{array}$$

  
  ::A=bh220=(b)(5)2

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  Begin isolating your unknown by multiplying both sides of the equation by 2.
  ::通过将方程的两边乘以2 开始孤立未知的你

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  $$\begin{array}{r}\begin{array}{rcl}20\times 2& =& \frac{(b)(5)}{2}\times 2\\ 40& =& 10b\end{array}\end{array}$$

  
  ::20x2=(b)(5)2x240=10b

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  Then, divide both sides by 10
  ::然后,将双方除以10

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  $$\begin{array}{r}4=b\end{array}$$

  
  ::4=b 4=b

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  The answer is 4 yards. Bill and Taken should stand about 4 yards apart from one another.
  ::答案是4码 比尔和乔德应该站在一起4码

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  Example 2
  ::例2

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  A triangle has a $\begin{array}{r}\text{base}=4\text{inches}\end{array}$  and an $\begin{array}{r}\text{area}=6\text{sq. inches}\end{array}$ . What is its height?
  ::三角形有一个基数=4英寸和一个区域=6 sq. 英寸。它的高度是多少?

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  First, write the formula and substitute the values that are given.
  ::首先,写入公式并替换给定的值。

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  $$\begin{array}{r}\begin{array}{rcl}A& =& \frac{bh}{2}\\ 6& =& \frac{4h}{2}\end{array}\end{array}$$

  
  ::A=bh26=4h2

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  Next, begin isolating your unknown by multiplying both sides of the equation by 2.
  ::接下来,通过将方程的两边乘以2 来孤立未知的你

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  $$\begin{array}{r}\begin{array}{rcl}6\times 2& =& \frac{4h}{2}\times 2\\ 12& =& 4h\end{array}\end{array}$$

  
  ::6x2=4h2x212=4h

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  Then, divide both sides by 4.
  ::然后,将双方除以4。

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  $$\begin{array}{r}3=h\end{array}$$

  
  ::3=h 3=h

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  The answer is 3 inches. The height of the triangle is 3 inches.
  ::答案是3英寸 三角形的高度是3英寸

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  Example 3
  ::例3

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  A triangle’s $\begin{array}{r}\text{height}=3\text{feet}\end{array}$ . Its $\begin{array}{r}\text{area}=7.5\text{sq. feet}\end{array}$ . What is the triangle’s base?
  ::三角形的高度=3英尺,面积=7.5平方米。三角形的底部是什么?

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  First, write the formula and substitute the values that are given.
  ::首先,写入公式并替换给定的值。

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  $$\begin{array}{r}\begin{array}{rcl}A& =& \frac{bh}{2}\\ 7.5& =& \frac{(b)(3)}{2}\end{array}\end{array}$$

  
  ::A=bh27.5=(b)(3)(3)2

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  Next, begin isolating your unknown by multiplying both sides of the equation by 2.
  ::接下来,通过将方程的两边乘以2 来孤立未知的你

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  $$\begin{array}{r}\begin{array}{rcl}7.5\times 2& =& \frac{(b)(3)}{2}\times 2\\ 15& =& 3b\end{array}\end{array}$$

  
  ::7.5x2=(b)(3)(3)2x215=3b

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  Then, divide both sides by 3.
  ::然后,将双方除以3。

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  $$\begin{array}{r}5=b\end{array}$$

  
  ::5=b 5=b

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  The answer is 5 feet. The base of the triangle is 5 feet.
  ::答案是5英尺 三角形的底部是5英尺

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  Example 4
  ::例4

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  A triangle’s $\begin{array}{r}\text{base}=7\text{meters}\end{array}$ , $\begin{array}{r}\text{area}=17.5sq.\end{array}$  meters, what is the height?
  ::三角形的基数=7米,面积=17.5平方米,高度是多少?

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  First, write the formula and substitute the values that are given.
  ::首先,写入公式并替换给定的值。

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  $$\begin{array}{r}\begin{array}{rcl}A& =& \frac{bh}{2}\\ 6& =& \frac{4h}{2}\end{array}\end{array}$$

  
  ::A=bh26=4h2

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  Next, begin isolating your unknown by multiplying both sides of the equation by 2.
  ::接下来,通过将方程的两边乘以2 来孤立未知的你

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  $$\begin{array}{r}\begin{array}{rcl}6\times 2& =& \frac{4h}{2}\times 2\\ 12& =& 4h\end{array}\end{array}$$

  
  ::6x2=4h2x212=4h

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  Then, divide both sides by 4.
  ::然后,将双方除以4。

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  $$\begin{array}{r}3=h\end{array}$$

  
  ::3=h 3=h

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  The answer is 3 inches. The height of the triangle is 3 inches.
  ::答案是3英寸 三角形的高度是3英寸

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  Example 5
  ::例5

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  A triangle has an area of 11.25 square feet and a base length of 4.5 ft. What is the height of the triangle?
  First, write the formula and substitute the values that are given.
  ::三角形的面积为11.25平方英尺,基长为4.5英尺。 三角形的高度是多少? 首先, 写入公式并替换给定的值 。

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  $$\begin{array}{r}\begin{array}{rcl}A& =& \frac{bh}{2}\\ 11.25& =& \frac{4.5h}{2}\end{array}\end{array}$$

  
  ::A=bh211.25=4.5h2

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  Next, begin isolating your unknown by multiplying both sides of the equation by 2.
  ::接下来,通过将方程的两边乘以2 来孤立未知的你

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  $$\begin{array}{r}\begin{array}{rcl}11.25\times 2& =& \frac{4.5h}{2}\times 2\\ 22.5& =& 4.5h\end{array}\end{array}$$

  
  ::11.25x2=4.5h2x222.5=4.5h

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  Then, divide both sides by 4.5
  ::然后,将双方除以4.5

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  $$\begin{array}{r}5=h\end{array}$$

  
  ::5=h 5=h

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  The answer is 5 feet. The height of the triangle is 5 feet.
  ::答案是5英尺 三角形的高度是5英尺

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  Review
  ::回顾

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  Find the missing base or height of the triangles given the area and one other dimension.
  ::查找给定区域和其他维度的三角形的缺失基数或高度。

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  1. Area = 13.5 sq. meters, Base = 9 meters
     ::面积=13.5平方米,基地=9米
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  2. Area = 21 sq. meters, Base = 7 meters
     ::面积=21平方米,基地=7米
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  3. Area = 12 sq. meters, Base = 8 meters
     ::面积=12平方米,基地=8米
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  4. Area = 33 sq. ft, Base = 11 feet
     ::=33平方英尺,基地=11英尺
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  5. Area = 37.5 sq. ft. Base = 15 feet
     ::基地=15英尺
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  6. Area = 60 sq. ft., height = 10 ft.
     ::面积=60平方英尺,高度=10平方英尺。
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  7. Area = 20.25 sq. in, height = 4.5 in
     ::面积=20.25平方米,高度=4.5平方米
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  8. Area = 72 sq. in, height = 8 in
     ::面积=72平方英寸,高度=8英寸
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  9. Area = 22.5 sq. feet, height = 5 feet
     ::面积=22.5平方英尺,高度=5英尺
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  10. Area = 12 sq. in, height = 4 in
      ::面积=12平方英寸,高度=4英寸
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  11. Area = 45 sq. in, height = 9 in
      ::面积=45平方英寸,高度=9英寸
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  12. Area = 84 sq. ft, height = 12 ft
      ::面积=84平方英尺,高度=12平方英尺
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  13. Area = 144 sq. in, height = 16 in
      ::面积=144平方英寸,高度=16英寸
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  14. Area = 144.5 sq. ft, height = 17 ft.
      ::面积=144.5平方英尺,高度=17平方英尺。
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  15. Area = 123.5 sq. in, height = 19 in
      ::面积=123.5平方米,高度=19平方米

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  Review (Answers)
  ::回顾(答复)

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  Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
  ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。

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  Resources
  ::资源